First pedal bike for little un!

I'd second the Specialized Hotrock 12" for size, quality and relative low weight. Got one for my daughter's third birthday (she is also on the diddy size) and it lasted her over two years before she outgrew it.
As per Allez the back pedal rear brake is a real PITA to explain to a 3 year old, a normal brake would be way better in my opinion.

Something small and cheap could do. Single speed is a really good idea as they're simple and your child won't want to ride far. The small Ridgebacks are popular with some of the families coming to our BMX club and those bikes are tough and straightforward. My sons started with a Raleigh from Halfords that was heavy but very tough and which was then handed on to friends for their kids. You could see what's for sale on ebay and screen those for ones that look suspicious or in poor condition. She'll grow out of it soon anyway so you don't need to spend that much. Just make sure it's safe to ride and not obviously stolen.

Cutting down an old MTB frame sounds like too much work.

Decathlon is worth a look, they can have a ride round the shop too.

My son graduated from an islabikes Rothan to a Cnoc - though think 14" wheels. The good thing about islabikes apart from the quality, weight and thought that goes into making them child friendly is the resale value. I expect to move our Rothan on in the next 4 months and hope to get a decent chunk of the purchase price back.

I'd start with the type of cycling you'd like to do with your kids and how strong they are. I've known families who have been very happy with superstore bikes but they've been pretty strong kids and they have confined their cycling to mucking about in the street outside their house.
If your kids are like mine, a bit weedy and/or you'd like to do longer rides it is worth trying to find lighter bikes with proper bearings (I once took a superstore "bottom bracket" apart - the pedal axle just rotated in plastic bushes). We have two Isla bikes and they've been a great success, my brother got his son a frog which he's also pleased with. Do pay attention to the size charts - I bought my eldest a bike that she was only just big enough for and she did struggle to begin with.

If you can find one, old Raleighs like this RSW11 are so much better than modern equivalents IMO.
Single speed
Easy to step through
Handlebars and seat go right down
...and retro-cool!

My twins (now 16), had them from about 4 years old and they were much better than the overweight contemporary bikes many of their friends had.

Islabikes fan here. Check the (accurate) size chart or give them a ring.

IME singlespeed bikes are fine for everything except steep hills. Gears are overrated for young children.

This recent thread might be of interest:
http://road.cc/content/forum/116628-kids-bikes

My two had Ridgeback MX14s and MX16s until they graduated on to Islabikes.

The Ridgebacks are hefty compared to the Islabikes but good compared to the catalogue type bikes that dominate bikes with this wheel size.

The Ridgebacks are pretty bomb proof - the MX16 is currently their 'stunt bike' and they try destroying it at local skate parks (with no success so far!)

Gears make cycling much more pleasurable, but small kids tend to destroy mech hangers and have little idea when to change gear. Both my kids (aged 6 & 7 now) race cyclocross and I have had to buy a gear alignment tool so common are prangs - and they are pretty good bike handlers (can do some red and black MTB skills).

Singlespeed bikes are only really suitable for flattish runs - e.g. along railway lines, and there weight doens't matter so much.

My two had a Specialized Hotrock 12"

The back pedal brake was a bit odd, took the internals out of it!

If its any help my son rode a (stabilizer-less) Ridgback MX12 from his 3rd birthday. Its heavier than islabikes, but we live in a flat area, so its not a big deal. The smallest islabike was too big for him at the time otherwise I would have bought that. An islabike will be his next bike.

The ridgback is well made, and has stood up to standard levels of abuse well. Brakes are ergonomically correct for him and easy to operate. There's probably another 6months-1years use in it before bigger wheels and gear inches would help. Will definitely be keeping it in the garage for the younger daughter.

Have you seen Islabikes?

I don't think they're *toooo* expensive (although perspective dependent, I don't have kids!) and friends who've bought them for their nippers have always raved about them - they seem to be a really nice, well thought out introduction to "proper" cycling for young people;

http://www.islabikes.co.uk/index-eu.html